ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Pages 80–87 (September 15, 2003) S 1079-6762(03)00114-8

ALGEBRAS OF PSEUDODIFFERENTIAL OPERATORS ON COMPLETE MANIFOLDS

BERND AMMANN, ROBERT LAUTER, AND VICTOR NISTOR

(Communicated by Michael E. Taylor)

Abstract. In several influential works, Melrose has studied examples of noncompact manifolds M_{0} whose large scale geometry is described by a Lie algebra of vector fields V ⊂ Γ(M; T M) on a compactification of M_{0} to a manifold with corners M. The geometry of these manifolds—called “manifolds with a Lie structure at infinity”—was studied from an axiomatic point of view in a previous paper of ours. In this paper, we define and study an algebra Ψ_{1,0,V}^{∞}(M_{0}) of pseudodifferential operators canonically associated to a manifold M_{0} with a Lie structure at infinity V ⊂ Γ(M; T M). We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to the algebras that we introduce. In particular, the algebra Ψ_{1,0,V}^{∞}(M_{0}) is a “microlocalization” of the algebra Diff_{V}^{∗}(M) of differential operators with smooth coefficients on M generated by V and C^{∞}(M). This proves a conjecture of Melrose (see his ICM 90 proceedings paper).

Introduction

In [17], Melrose has formulated a far reaching program to study the analytic properties of geometric differential operators on an open manifold M_{0}, provided that its large scale geometry is controlled by a Lie algebra of vector fields V on a compactification M ⊃ M_{0}. Typically, M is a manifold with corners, M_{0} = M ∖∂M is obtained by removing all faces of M, and the given Lie algebra of vector fields V ⊂ Γ(M; T M) satisfies a number of axioms (see Section 1). This structure leads to complete metrics of bounded curvature on M_{0}, and ∂M is the “boundary at infinity.” For example, manifolds with asymptotically Euclidean, asymptotically hyperbolic, or asymptotically complex hyperbolic ends are obtained in this way.

An important ingredient in Melrose’s program mentioned above is to define a suitable pseudodifferential calculus on M_{0} adapted in a certain sense to (M, V). Melrose calls this pseudodifferential calculus a “microlocalization of Diff_{V}^{∗}(M),” where Diff_{V}^{∗}(M) is the algebra of differential operators on M generated by V and C^{∞}(M).

Received by the editors April 24, 2003. 2000 Mathematics Subject Classification. Primary 58J40; Secondary 58H05, 65R20. Key words and phrases. Differential operator, pseudodifferential operator, principal symbol,

conormal distribution, Riemannian manifold, Lie algebra, exponential map.

Ammann was partially supported by the European Contract Human Potential Program, Research Training Networks HPRN-CT-2000-00101 and HPRN-CT-1999-00118; Nistor was partially supported by NSF Grants DMS 99-1981 and DMS 02-00808. Manuscripts available from

http://www.math.psu.edu/nistor/.

\begin{matrix}c \\ ◯ \end{matrix}2003 American Mathematical Society

In [19] and several other papers, Melrose and his collaborators have completed this program in several important cases [6, 9, 10, 13, 14, 15, 16, 18, 19, 20, 21, 22, 32]. One of the main points is that the geometric operators on manifolds with a Lie structure at infinity identify with degenerate differential operators on the compactification M. This type of differential operators appear naturally, for example, in the study of boundary value problems on manifolds with singularities. Results in this direction were obtained also by Schulze and his collaborators, who typically worked in the framework of the Boutet de Monvel algebras. See [26, 27] and the references therein. See also [11, 24, 25].

It is desirable to present all these cases in a unified setting and to extend the results to a larger class of manifolds, namely the class of “manifolds with a Lie structure at infinity.” These are open manifolds M_{0} which are topologically the interior of a compact manifold M with corners, and the geometry of M_{0} near ∂M is described by a Lie algebra of vector fields V ⊂ Γ(T M) satisfying certain axioms (see Section 1 for details). The geometrical properties of these manifolds were studied in [1]. Here we introduce an algebra Ψ_{1,0,V}^{∞}(M_{0}) of pseudodifferential operators on M_{0} that is canonically associated to the manifold with a Lie structure at infinity M_{0}. Then we show that this algebra “microlocalizes” Diff_{V}^{∗}(M) in the sense that Diff_{V}^{∗}(M) is the algebra of all differential operators in Ψ_{1,0,V}^{∞}(M_{0}) and that Ψ_{1,0,V}^{∞}(M_{0}) has the usual symbolic properties of algebras of pseudodifferential operators on a compact manifold. We also show that the algebra Ψ_{1,0,V}^{∞}(M_{0}) is invariant under the diffeomorphisms of M_{0} obtained by exponentiating the vector fields X ∈ V and under conjugation with complex powers of the functions that define the faces of the compactification M of M_{0}.

The explicit construction of the algebra Ψ_{1,0,V}^{∞}(M_{0}) microlocalizing Diff_{V}^{∗}(M_{0}) in the sense of [17] is, roughly, as follows. First, V defines an extension of T M_{0} to a vector bundle A → M (M_{0} = M ∖ ∂M). Denote V_{r} := {d(x, y) < r} ⊂ M_{0}^{2} and (A)_{r} = {v ∈ A, ∥v∥ < r}. Let r > 0 be less than the injectivity radius of M_{0} and V_{r} ∋ (x, y) ↦ (x, τ(x, y)) ∈ (A)_{r} be a local inverse of the Riemannian exponential map T M_{0} ∋ v ↦ exp_{x}(−v) ∈ M_{0} × M_{0}. Let χ be a smooth function on A with support in (A)_{r}, χ = 1 on (A)_{r/2}. For any a ∈ S_{1,0}^{m}(A^{∗}), we define

(1) [a_{χ}(D)u](x) = (2π)^{−n} ∫_{M_{0}} (∫_{T_{x}^{∗}M_{0}} e^{iτ(x,y)·η}χ(x, τ(x, y))a(x, η)u(y) dη) dy.

The algebra Ψ_{1,0,V}^{∞}(M_{0}) is then generated linearly by the operators a_{χ}(D) and b_{χ}(D) exp(X_{1}) … exp(X_{k}), a ∈ S^{∞}(A^{∗}), b ∈ S^{−∞}(A^{∗}), and X_{j} ∈ V. (We need to introduce the operators b_{χ}(D) exp(X_{1}) … exp(X_{k}), where exp(X_{j}) is the exponential of the vector field X_{j}, to make our space of operators closed under products.) The definition of the operators of order −∞ is in fact the crucial part of this construction.

In [17], Melrose outlined the construction of a pseudodifferential calculus on manifolds with Lie structure at infinity provided certain manifolds with corners (blow-ups) can be constructed. (See also [31].) Our approach owes a lot to his approach when it comes to proving that Ψ_{1,0,V}^{∞}(M_{0}) are algebras, but we replace the blow-ups with groupoids, using also a deep result of Crainic and Fernandes [4] on the integration of Lie algebroids. (However, to prove the original form of the conjecture from [17], the earlier results of [23] also suffice.)

This paper is an announcement. Complete proofs will be published in [2].

Acknowledgements: V. N. would like to thank the Institute Erwin Schrödinger in Vienna, where part of this work was completed. We also thank A. Vasy who has contributed in several ways to the results in this paper.

1. Manifolds with Lie structure at infinity

For the convenience of the reader, let us recall the definition of Riemannian manifolds with a Lie structure at infinity and some of their basic properties [1].

In the sequel, by a manifold we shall always understand a C^{∞}-manifold with corners, whereas a smooth manifold is a C^{∞}-manifold without corners. By definition, every point p in a manifold with corners M has a coordinate neighborhood diffeomorphic to [0, ∞)^{k} × ℝ^{n−k} such that the transition functions are smooth up to the boundary. We then call p a point of boundary depth at most k and write codim(p) ≤ k. Points p with codim(p) ≤ k but not codim(p) ≤ k − 1 are said to be of boundary depth k. (This terminology is in agreement with the terminology for stratified spaces, if we stratify a manifold with corners by its open subfaces; see [12].) We denote by ∂M the union of all non-trivial faces of M. Usually, we write M_{0} for the interior of M, i.e., M_{0} := M ∖ ∂M.

A Lie subalgebra V ⊆ Γ(M; T M) of the Lie algebra of all smooth vector fields on M is said to be a structural Lie algebra of vector fields provided it is a finitely generated, projective C^{∞}(M)-module and each V ∈ V is tangent to all hyperfaces of M. (We shall denote the sections of a vector bundle V → X by Γ(X; V), unless X is understood, in which case we shall write simply Γ(V).) By the Serre-Swan theorem [8], there exists a smooth vector bundle A = A_{V} → M together with a natural map ϱ = ϱ_{V} : A → T M making the following diagram commutative

(2)  \begin{aligned}A_{V} & & ⟶^{ϱ} & & T M \\ & ↘ & & ↙ \\ & & M \end{aligned}

and such that V = ϱ(Γ(A_{V})).

Definition 1.1. A Lie structure at infinity on a smooth manifold M_{0} is a triple (M_{0}, M, V), where M is a compact manifold, possibly with corners, and V ⊂ Γ(M; T M) is a structural Lie algebra of vector fields on M with the following properties:

(a) M_{0} is diffeomorphic to the interior M ∖ ∂M of M;

(b) If ϱ : A → T M is the anchor map defined by diagram (2), then ϱ restricts to an isomorphism A|_{M∖∂M} → T M|_{M∖∂M}.

Note that for a given manifold M_{0} in general there can exist many Lie structures at infinity. Examples of Lie structures at infinity were discussed in [1].

From now on, we identify M_{0} with M ∖∂M and A|_{M_{0}} with T M_{0}. Because A and V determine each other up to isomorphism, we sometimes specify a Lie structure at infinity on M by the pair (M, A). Also, it follows that ϱ : Γ(M; A) → Γ(M; T M) is injective, so we shall identify Γ(M; A) with V = ϱ(Γ(M; A)).

Elements in the enveloping algebra Diff_{V}^{∗}(M) of V are called V-differential operators on M. By the injectivity of the induced structural map ϱ_{V} : Γ(A_{V}) → Γ(T M), the algebra of V-differential operators can be realized as a subalgebra of all differential operators on M; in particular, they act continuously on the space C^{∞}(M). Moreover, the order of differential operators induces a filtration Diff_{V}^{m}(M), m ∈ ℤ_{+}, on the algebra Diff_{V}^{∗}(M). Since Diff_{V}^{∗}(M) is a C^{∞}(M)-module, we can introduce Vdifferential operators acting between sections of smooth vector bundles E, F → M, E, F ⊂ M × ℂ^{N} by

(3)  Diff_{V}^{∗}(M; E, F) := e_{F} M_{N}(Diff_{V}^{∗}(M))e_{E} ,

where e_{E}, e_{F} ∈ M_{N}(C^{∞}(M)) are the projections onto E and, respectively, F. It follows that Diff_{V}^{∗}(M; E, E) =: Diff_{V}^{∗}(M; E) is an algebra that is closed under adjoints and contains all geometric operators on M_{0} that are associated to a metric on M_{0} that comes from a metric on A. (See [1].)

Since any metric on A induces a natural metric on T M_{0} = A|_{M_{0}}, we obtain the following definition.

Definition 1.2. A manifold M_{0} with a Lie structure at infinity (M, A) and with metric g_{0} on M_{0} obtained by restricting a metric g from A to T M_{0} is called a Riemannian manifold with a Lie structure at infinity.

The geometry of Riemannian manifolds (M_{0}, g_{0}) with a Lie structure at infinity has been studied in [1]. For instance, (M_{0}, g_{0}) is necessarily of infinite volume and complete. Moreover, all the covariant derivatives of the Riemannian curvature tensor are bounded. Under additional mild assumptions, we also know that the injectivity radius injrad(p), viewed as a function depending on p ∈ M, is bounded below by a positive constant, or, equivalently, (M_{0}, g_{0}) is of bounded geometry in the sense of [28] and references therein. We shall denote by r_{0} := inf_{p} injrad(p) the injectivity radius of M_{0}.

On a Riemannian manifold M_{0} with a uniform structure at infinity (M, A), the exponential map exp_{p} : T M_{0} → M_{0} is well defined for all p ∈ M_{0} and extends to a differentiable map exp_{p} : A_{p} → M depending smoothly on p ∈ M. A convenient way to introduce the exponential map is via the geodesic spray, as done in [1]. Similarly, any vector field X ∈ Γ(A) is integrable and will map any (connected) face to itself. The resulting diffeomorphism of M_{0} will be denoted ψ_{X}.

We shall also assume from now on that r_{0}, the injectivity radius of (M_{0}, g_{0}), is positive.

2. Kohn-Nirenberg quantization and pseudodifferential operators

We now introduce the algebras Ψ_{cl,V}(M_{0}) and Ψ_{1,0,V}(M_{0}) of pseudodifferential operators on M_{0} adapted to the Lie structure at infinity (M, V). We also state some of their main properties.

2.1. Riemann-Weyl fibration. Fix now a Riemannian metric g on the bundle A, and let g_{0} = g|_{M_{0}} be its restriction to the interior M_{0} of M, defined in view of the identification A|_{M_{0}} = T M_{0}. We shall use this metric to trivialize all density bundles on M. Denote by π : T M_{0} → M_{0} the natural projection.

Define

(4)  Φ : T M_{0} ⟶ M_{0} × M_{0}, Φ(v) := (x, exp_{x}(−v)), x = π(v).

Recall that, for v ∈ T_{x}M, we have exp_{x}(v) = γ_{v}(1), where γ_{v} is the unique geodesic with γ_{v}(0) = π(v) = x and γ′_{v}(0) = v. It is known that there is an open neighborhood U of the zero-section M_{0} in T M_{0} such that Φ|_{U} is a diffeomorphism onto an open neighborhood V of the diagonal Δ_{M_{0}} ⊆ M_{0} × M_{0}.

To fix notation, let E be a vector bundle with a norm ∥ · ∥. We shall denote by (E)_{r} the set of all vectors v of E with norm ∥v∥ < r. Our assumption that the injectivity radius r_{0} of M_{0} is positive gives that, for 0 < r ≤ r_{0}, the restriction Φ|_{(T M_{0})_{r}} is a diffeomorphism onto an open neighborhood V_{r} of the diagonal Δ_{M_{0}}. We continue, by slight abuse of notation, to write Φ for that restriction. Following [5], we shall call Φ a Riemann-Weyl fibration. However, note that the Riemann-Weyl fibrations are defined in a slightly different way in [5]; the difference will be of no importance for us. The inverse of Φ is given by

M_{0} × M_{0} ⊇ V_{r} ∋ (x, y) ⟼ (x, τ(x, y)) ∈ (T M_{0})_{r} ,

where τ(x, y) ∈ T_{x}M_{0} is the tangent vector at x to the shortest geodesic γ : [0, 1] → M such that γ(0) = x and γ(1) = y.

We shall denote by S_{1,0}^{m}(E) the space of symbols of order m and type (1, 0) on E (in Hörmander’s sense) and by S_{cl}^{m}(E) the space of classical symbols of order m on E [7, 29, 30]. These spaces are reviewed in [2] in our framework.

Let χ ∈ C^{∞}(A^{∗}) be a smooth function that is equal to 1 on (A^{∗})_{r} and is equal to 0 outside (A^{∗})_{2r}, for some r < r_{0}/3. Then, following [2], we define

a_{χ}(D)u(x) = (2π)^{−n} ∫_{T^{∗}M_{0}} e^{iτ(x,y)·η}χ(x, τ(x, y))a(x, η)u(y) dη dy .

This integral is an oscillatory integral with respect to the symplectic measure on T^{∗}M_{0} [7]. Alternatively, we can consider the measures on M_{0} and on T_{x}^{∗}M_{0} defined by some choice of a metric on A and then we integrate first along the fibers T_{x}^{∗}M_{0} and then along M_{0}.

Proposition 2.1. The map σ_{tot} : S_{1,0}^{m}(A^{∗}) → Ψ^{m}(M_{0})/Ψ^{−∞}(M_{0}),

σ_{tot}(a) := a_{χ}(D) + Ψ^{−∞}(M_{0}),

is independent of the choice of the function χ ∈ C_{c}^{∞}((A)_{r}) used to define a_{χ}(D).

The space of all operators of the form a_{χ}(D), with a ∈ S_{1,0}^{∗}(A^{∗}), is not closed under composition. In order to make it closed under composition, we are going to include more operators of order −∞ in our calculus.

Any vector field X ∈ Γ(A) generates a global flow Ψ_{X} : ℝ × M → M because X is tangent to all boundary faces of M and M is compact. Evaluating at t = 1 yields a diffeomorphism

(5)  ψ_{X} := Ψ_{X}(1, ·) : M → M.

We continue to assume that the injectivity radius r_{0} of our fixed manifold with a Lie structure at infinity (M, M_{0}, A) is strictly positive.

Definition 2.2. Fix 0 < r < r_{0} and χ ∈ C_{c}^{∞}((A)_{r}) such that χ = 1 in a neighborhood of M in A. For m ∈ ℝ, the space Ψ_{1,0,V}^{m}(M_{0}) of pseudodifferential operators generated by the Lie structure at infinity (M, A) is the linear space of operators C_{c}^{∞}(M_{0}) → C_{c}^{∞}(M_{0}) generated by a_{χ}(D), a ∈ S_{1,0}^{m}(A^{∗}), and b_{χ}(D)ψ_{X_{1}} … ψ_{X_{k}}, b ∈ S^{−∞}(A^{∗}) and X_{j} ∈ Γ(A), for all j.

Similarly, the space Ψ_{cl,V}^{m}(M_{0}) of classical pseudodifferential operators generated by the Lie structure at infinity (M, A) is obtained by using classical symbols a in the construction above.

We also obtain that Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,V}^{m}(M_{0}) are algebras independent of the choices made in their definition.

Theorem 2.3. The spaces Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,V}^{m}(M_{0}) are filtered algebras. They do not depend on the choice of the metric on A and the function χ used to define it, but depend, in general, on the Lie structure at infinity (M, A) on M_{0}.

The fact that Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,V}^{m}(M_{0}) are filtered algebras means that

Ψ_{1,0,V}^{m}(M_{0})Ψ_{1,0,V}^{m′}(M_{0}) ⊆ Ψ_{1,0,V}^{m+m′}(M_{0}) , Ψ_{cl,V}^{m}(M_{0})Ψ_{cl,V}^{m′}(M_{0}) ⊆ Ψ_{cl,V}^{m+m′}(M_{0}) ,

for all m, m′ ∈ ℂ ∪ {−∞}.

The proof of Theorem 2.3 is obtained by realizing Ψ_{1,0,V}^{∞}(M_{0}) as the homomorphic image of the algebra Ψ_{1,0}^{∞}(G) of pseudodifferential operators on a groupoid G integrating the Lie algebroid A (that is, with Γ(A) = V). This is possible due to the results of [4, 23, 24]. Note that this proof provides also an alternative definition of the algebras Ψ_{1,0,V}^{∞}(M_{0}) and Ψ_{cl,V}^{∞}(M_{0}). The advantage of our original definition, however, is that it is intrinsically formulated in terms of the geometry of M.

As for the usual algebras of pseudodifferential operators, we have the following basic property of the principal symbol.

Proposition 2.4. The principal symbol establishes isomorphisms

(6)  σ^{(m)} : Ψ_{1,0,V}^{m}(M_{0})/Ψ_{1,0,V}^{m−1}(M_{0}) → S_{1,0}^{m}(A^{∗})/S_{1,0}^{m−1}(A^{∗})

and

(7)  σ^{(m)} : Ψ_{cl,V}^{m}(M_{0})/Ψ_{cl,V}^{m−1}(M_{0}) → S_{cl}^{m}(A^{∗})/S_{cl}^{m−1}(A^{∗}) ≃ C^{∞}(S^{∗}A),

where S^{∗}A is the set of unit vectors in A^{∗}.

We have the following boundedness result.

Proposition 2.5. Any operator P ∈ Ψ_{1,0,V}^{m}(M_{0}) defines a continuous linear operator on C_{c}^{∞}(M_{0}) and C^{∞}(M). If m = 0, it also defines a bounded operator on L^{2}(M_{0}).

Part (i) of the following result is an analog of a standard result about the bcalculus [18], whereas the second formula is the independence of diffeomorphisms of the algebras Ψ_{cl,V}^{∞}(M_{0}), in the framework of manifolds with a Lie structure at infinity. Recall that if X ∈ Γ(A), we have denoted by ψ_{X} := Ψ_{X}(1, ·) : M → M the diffeomorphism defined by integrating X (and specializing at t = 1).

Proposition 2.6. (i) Let x be a defining function of some hyperface of M. Then

x^{s}Ψ_{1,0,V}^{m}(M_{0})x^{−s} = Ψ_{1,0,V}^{m}(M_{0}) and x^{s}Ψ_{cl,V}^{m}(M_{0})x^{−s} = Ψ_{cl,V}^{m}(M_{0})

for any s ∈ ℂ.

(ii) Similarly,

ψ_{X}Ψ_{1,0,V}^{m}(M_{0})ψ_{X}^{−1} = Ψ_{1,0,V}^{m}(M_{0}) and ψ_{X}Ψ_{cl,V}^{m}(M_{0})ψ_{X}^{−1} = Ψ_{cl,V}^{m}(M_{0}),

for any X ∈ Γ(A).

Let us notice that (ii) remains true for any diffeomorphism of M_{0} that extends to an automorphism of (M, A). Recall that an autormorphism of the Lie algebroid (M, A) is a morphism of vector bundles (ϕ, ψ), ϕ : M → M, ψ : A → A, such that ϕ and ψ are diffeomorphisms and

ϱ_{Γ} ∘ ψ_{Γ} = ϕ_{∗} ∘ ϱ_{Γ},

where ϱ_{Γ} : Γ(A) → Γ(T M) and ψ_{Γ} : Γ(A) → Γ(A) are the maps defined by the anchor map ϱ and ψ, respectively, and ϕ_{∗} : Γ(T M) → Γ(T M) is given by the differential of ϕ.

A proof of the above proposition can be obtained using the corresponding results for the algebras of pseudodifferential operators on G.

The proof of the following proposition relies on the Campbell-Hausdorff formula.

Proposition 2.7. Let X ∈ Γ(A) and denote by a_{X}(ξ) = ξ(X) the associated linear function on A^{∗}. Then a_{X} ∈ S^{1}(A^{∗}) and a_{X}(D) = −ıX. Moreover,

{a_{χ}(D), a = polynomial in each fiber } = Diff_{V}^{∗}(M_{0}).

From this we obtain the following result.

Theorem 2.8. Let Diff(M_{0}) be the algebra of all differential operators on M_{0}. Then

Ψ_{1,0,V}^{∞}(M_{0}) ∩ Diff(M_{0}) = Diff_{V}^{∗}(M_{0}).

Since Diff_{V}^{∗}(M_{0}) ⊂ Ψ_{cl,V}^{∞}(M_{0}), it also follows that

Ψ_{cl,V}^{∞}(M_{0}) ∩ Diff(M_{0}) = Diff_{V}^{∗}(M_{0}).

Thus the algebras Ψ_{1,0,V}^{∞}(M_{0}) and Ψ_{1,0,V}^{∞}(M_{0}) are microlocalizations of Diff_{V}^{∗}(M_{0}), which, together with the other properties of these algebras stated above, shows that our constructions solve a conjecture from [17].

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Universität Hamburg, Fachbereich 11–Mathematik, Bundesstrasse 55, D-20146 Hamburg, Germany

E-mail address: ammann@berndammann.de

Universität Mainz, Fachbereich 17–Mathematik, D-55099 Mainz, Germany

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Mathematics Department, Pennsylvania State University, University Park, PA 16802

E-mail address: nistor@math.psu.edu